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Grid Points on Hyperbolas

Find a condition which determines whether the hyperbola y^2 - x^2 = k contains any points with integer coordinates.

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Here is a pattern for you to experiment with using graph drawing software. Find the equations of the graphs in the pattern.

Lattice Points

Age 16 to 18 Challenge Level:

(1) Show that if there is one lattice point (point with integer coordinates) on the parabola $$y=ax^2$$ then there are infinitely many.

(2) Find all the lattice points on the hyperbola $$x^2 - y^2 = 84.$$